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Can quantum walks provide practical exponential speedups

B. L. Douglas, Jingbo B. Wang

Published Jun 3, 2007• 1 citation
OpenAlex

Abstract

Quantum walks, being the quantum analogue of classical random walks, are expected to provide a fruitful source of quantum algorithms. A few such algorithms have already been developed, including the `glued trees' algorithm, which provides an exponential speedup over classical methods, relative to a particular quantum oracle. Here, we discuss the possibility of a quantum walk algorithm yielding such an exponential speedup over possible classical algorithms, without the use of an oracle. We provide examples of some highly symmetric graphs on which efficient quantum circuits implementing quantum walks can be constructed, and discuss potential applications to quantum search for marked vertices along these graphs.

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Quantum walks, being the quantum analogue of classical random walks, are expected to provide a fruitful source of quantum algorithms. A few such algorithms have already been developed, including the `glued trees' algorithm, which provides an exponential speedup over classical methods, relative to a particular quantum oracle. Here, we discuss the possibility of a quantum walk algorithm yielding such an exponential speedup over possible classical algorithms, without the use of an oracle. We provide examples of some highly symmetric graphs on which efficient quantum circuits implementing quantum walks can be constructed, and discuss potential applications to quantum search for marked vertices along these graphs.
Quantum walks, being the quantum analog of classical random walks, are expected to provide a fruitful source of quantum algorithms. A few such algorithms have already been developed, including the ``glued trees'' algorithm, which provides an exponential speedup over classical methods, relative to a particular quantum oracle. Here, we discuss the possibility of a quantum walk algorithm yielding such an exponential speedup over possible classical algorithms, without the use of an oracle. We provide examples of some highly symmetric graphs on which efficient quantum circuits implementing quantum walks can be constructed and discuss potential applications to quantum search for marked vertices along these graphs.
Quantum random walks on graphs have been shown to display many interesting properties, including exponentially fast hitting times when compared with their classical counterparts. However, it is still unclear how to use these novel properties to gain an algorithmic speedup over classical algorithms. In this paper, we present a quantum search algorithm based on the quantum random-walk architecture that provides such a speedup. It will be shown that this algorithm performs an oracle search on a database of N items with O(N)O(\sqrt{N})O(N​) calls to the oracle, yielding a speedup similar to other quantum search algorithms. It appears that the quantum random-walk formulation has considerable flexibility, presenting interesting opportunities for development of other, possibly novel quantum algorithms.
Abstract There are few known exponential speedups for quantum algorithms and these tend to fall into even fewer families. One speedup that has mostly resisted generalization is the use of quantum walks to traverse the welded-tree graph, due to Childs, Cleve, Deotto, Farhi, Gutmann, and Spielman. We show how to generalize this to a large class of hierarchical graphs in which the vertices are grouped into “supervertices” which are arranged according to a d-dimensional lattice. Supervertices can have different sizes, and edges between supervertices correspond to random connections between their constituent vertices. The hitting times of quantum walks on these graphs are related to the localization properties of zero modes in certain disordered tight binding Hamiltonians. The speedups range from superpolynomial to exponential, depending on the underlying dimension and the random graph model. We also provide concrete realizations of these hierarchical graphs, and introduce a general method for constructing graphs with efficient quantum traversal times using graph sparsification.
This paper presents a set of highly efficient quantum circuits for discrete-time quantum walks on non-degree-regular graphs. In particular, we describe a general procedure for constructing highly efficient quantum circuits for quantum walks on star graphs of any degree and Cayley trees with an arbitrary number of layers, which are nonsparse in general. We also show how to modify these circuits to implement a full quantum-walk search algorithm on these graphs, without reference to a ``black-box'' oracle. This provides a practically implementable method to explore quantum-walk-based algorithms with the aim of eventual real-world applications.
We present several families of graphs that allow both efficient quantum walk implementations and efficient quantum walk based search algorithms. For these graphs, we construct quantum circuits that explicitly implement the full quantum walk search algorithm, without reference to a `black box' oracle. These circuits provide a practically implementable method to explore quantum walk based search algorithms with the aim of eventual real-world applications. We also provide a numerical analysis of a quantum walk based search along a twisted toroid family of graphs, which requires O(n\sqrt{n}n​ log(nnn)) elementary 2-qubit quantum gate operations to find a marked node.
This paper investigates the application of quantum walk algorithms to large-scale graph search problems. Classical algorithms often struggle with the computational complexity of exploring vast graphs, particularly when searching for specific patterns or nodes. Quantum walks, inspired by the probabilistic nature of random walks, provide a promising alternative by utilizing quantum superposition and interference to accelerate the search process. We explore the theoretical framework and potential advantages of employing quantum walks for graph traversal, focusing on scenarios where exponential speedups are theoretically possible. This work aims to demonstrate the potential of quantum computing to tackle computationally demanding graph search tasks that are currently intractable for classical methods. The core concept leverages the ability to simultaneously explore multiple paths within a graph, drastically reducing the search time compared to sequential exploration. We present a foundational understanding of quantum walk implementation and analyze its suitability for various graph search algorithms.